Science:Math Exam Resources/Courses/MATH152/April 2011/Question A 22

MATH152 April 2011

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Question A 22
Questions A22-A24 concern a random walk with three states that has transition matrix $\left[{\begin{array}{c c c}8/10&3/10&1/10\\2/10&3/10&7/10\\0&4/10&2/10\end{array}}\right]$ If the walker is equally likely to start in each of the three states, what is the chance that she will be in state 3 at the next time step?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall that a transition matrix with element i,j (row i and column j) is the probability of reaching state i coming from state j. The vector input is the initial states and the vector output is the final states.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If the transition matrix is $P={\begin{bmatrix}8/10&3/10&1/10\\2/10&3/10&7/10\\0&4/10&2/10\end{bmatrix}}$ then the operation Px=b gives us the probability of being in each of the three states b based on where we started, x. We are told that we are equally likely to be in any of the three states so we have as our vector, $\mathbf {x} =\left[1/3,1/3,1/3\right].$ To find out the probabilities of being in each state after one transition we compute Px, $P\mathbf {x} ={\begin{bmatrix}8/10&3/10&1/10\\2/10&3/10&7/10\\0&4/10&2/10\end{bmatrix}}{\begin{bmatrix}1/3\\1/3\\1/3\end{bmatrix}}={\begin{bmatrix}2/5\\2/5\\1/5\end{bmatrix}}.$ Therefore we get that the probability of being in state 3 is 1/5. Notice that the output probabilities sum to 1. This is a requirement since we only allow transitions from states 1 to 3, this summation to 1 simply says that no matter where we start we will end up somewhere in states 1 to 3.